Algebra of Sets (Summary of Laws) –
Sets under the operations of union, intersection, and complements, satisfy the laws listed below. You can verify these laws by drawing Venn Diagrams.
If A, B, C are any sets and ξ is the universal set, then –
1. (A’)’ = A
2. ϕ’ = ξ
=> ξ’ = ϕ
3. A ∪ ϕ = A
=> A ∩ ξ = A
4. A ∪ ξ = ξ (Indempotent Laws)
=> A ∩ ϕ = ϕ
5. A ∪ A = A
=> A ∩ A = A
6. A ∪ A’ = ξ
=> A ∩ A’ = ϕ
7. (A ∪ B) ∪ C = A ∪ (B ∪ C) [Associative Laws]
=> (A ∩ B) ∩ C = A ∩ (B ∩ C)
8. A ∪ B = B ∪ A [Commutative Laws]
=> A ∩ B = B ∩ A
9. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) [Distributive Laws]
=> A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Union & Intersection are distributive over intersection and union respectively.
10. (A ∪ B)’ = A’ ∩ B’ [De Morgan’s Laws]
=> (A ∩ B)’ = A’ ∪ B’
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